Question
Factorize the following polynomials: $x^3 + 13x^2 + 31x - 45$ given that $x + 9$ is a factor.
✓
Answer
Let $f(x) = x^3 + 13x^2 + 31x - 45$ be the given polynomial.
Therefore, $(x + 9)$ is a factor of the polynomial $f(x).$
Now, $f(x) = x^2(x + 9) + 4x(x + 9) - 5(x + 9)$
$= (x + 9)(x^2 + 4x - 5)$
$= (x + 9)(x^2 +5x - x - 5)$
$= (x + 9)(x - 1)(x + 5)$
Hence $(x - 1), (x + 5)$ and $(x + 9)$ are the factors of polynomial $f(x).$
Therefore, $(x + 9)$ is a factor of the polynomial $f(x).$
Now, $f(x) = x^2(x + 9) + 4x(x + 9) - 5(x + 9)$
$= (x + 9)(x^2 + 4x - 5)$
$= (x + 9)(x^2 +5x - x - 5)$
$= (x + 9)(x - 1)(x + 5)$
Hence $(x - 1), (x + 5)$ and $(x + 9)$ are the factors of polynomial $f(x).$
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